| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper algebraic form then partial fractions |
| Difficulty | Standard +0.3 This is a standard partial fractions question with improper form requiring polynomial division first, followed by routine integration. The working involves familiar techniques (repeated linear factors in denominator) and straightforward logarithm/power rule integration, making it slightly easier than average but still requiring multiple steps and careful algebra. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply the form \(\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+2}\) | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(A=-1\), \(B=3\), \(C=2\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | Allow in the form \(\frac{Ax+B}{x^2}+\frac{C}{x+2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate and obtain terms \(\ln x - \frac{3}{x}+2\ln(x+2)\) | B1FT + B1FT + B1FT | The FT is on \(A\), \(B\), \(C\); or on \(A\), \(D\), \(E\) |
| Substitute limits correctly in an integral with terms \(a\ln x\), \(\frac{b}{x}\) and \(c\ln(x+2)\), where \(abc\neq 0\) | M1 | \(-\ln4-\frac{3}{4}+2\ln6(+\ln1)+3-2\ln3\) |
| Obtain \(\frac{9}{4}\) following full and exact working | A1 | AG – work to combine or simplify logs is required |
## Question 8(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply the form $\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+2}$ | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of $A=-1$, $B=3$, $C=2$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | Allow in the form $\frac{Ax+B}{x^2}+\frac{C}{x+2}$ |
## Question 8(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate and obtain terms $\ln x - \frac{3}{x}+2\ln(x+2)$ | B1FT + B1FT + B1FT | The FT is on $A$, $B$, $C$; or on $A$, $D$, $E$ |
| Substitute limits correctly in an integral with terms $a\ln x$, $\frac{b}{x}$ and $c\ln(x+2)$, where $abc\neq 0$ | M1 | $-\ln4-\frac{3}{4}+2\ln6(+\ln1)+3-2\ln3$ |
| Obtain $\frac{9}{4}$ following full and exact working | A1 | AG – work to combine or simplify logs is required |8 Let $\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 6 } { x ^ { 2 } ( x + 2 ) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence, showing full working, show that the exact value of $\int _ { 1 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x$ is $\frac { 9 } { 4 }$.\\
\hfill \mbox{\textit{CAIE P3 2019 Q8 [10]}}